scholarly journals Locating-Dominating Sets and Identifying Codes in Graphs of Girth at least 5

10.37236/4562 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Camino Balbuena ◽  
Florent Foucaud ◽  
Adriana Hansberg
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Upper Bounds ◽  
Disjoint Paths ◽  
Minimum Size ◽  
Dominating Sets ◽  
Identifying Codes ◽  
Identifying Code ◽  
Codes In Graphs ◽  
Vertex Disjoint

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meeting these bounds.

Paths, Stars and the Number Three

1996 ◽  
Vol 5 (3) ◽  
pp. 277-295 ◽  
Author(s):  
Bruce Reed
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Disjoint Paths ◽  
Cubic Graph ◽  
Vertex Disjoint

A dominating set for a graph G is a set D of vertices of G such that every vertex of G not in D is adjacent to a vertex of D. We prove that any graph G of minimum degree at least three contains a dominating set D of size at most 3|V(G)|/8. A star S is a graph consisting of a centre x and a set of edges from x to S — x. Clearly, a dominating set D for a graph G corresponds to a set of |D| stars which cover V(G). Thus, we show that the vertices of any graph G of minimum degree 3 can be covered by at most 3|V(G)|/8 vertex disjoint stars. We also show that any connected cubic graph G can be covered by [|V(G)|/9] vertex disjoint paths. Both these results are sharp.

scholarly journals Optimal Identifying Codes in Oriented Paths and Circuits

2021 ◽  
Vol 15 ◽  
pp. 9-14
Author(s):  
Ahmed Semri ◽  
Hillal Touati
Keyword(s):  
Sensor Networks ◽  
Connected Graph ◽  
Multiprocessor Systems ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Identifying Codes ◽  
Identifying Code ◽  
Faulty Processor ◽  
Classical Notion ◽  
Codes In Graphs

Identifying codes in graphs are related to the classical notion of dominating sets [1]. Since there first introduction in 1998 [2], they have been widely studied and extended to several application, such as: detection of faulty processor in multiprocessor systems, locating danger or threats in sensor networks. Let G=(V,E) an unoriented connected graph. The minimum identifying code in graphs is the smallest subset of vertices C, such that every vertex in V have a unique set of neighbors in C. In our work, we focus on finding minimum cardinality of an identifying code in oriented paths and circuits

scholarly journals Domination of the Rectangular Queen's Graph

10.37236/6026 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Sándor Bozóki ◽  
Péter Gál ◽  
István Marosi ◽  
William D. Weakley
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Minimum Size ◽  
Dominating Sets ◽  
Online Appendix ◽  
Independent Domination ◽  
Independent Dominating Set

The queens graph $Q_{m \times n}$ has the squares of the $m \times n$ chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal of the board. A set $D$ of squares of $Q_{m \times n}$ is a dominating set for $Q_{m \times n}$ if every square of $Q_{m \times n}$ is either in $D$ or adjacent to a square in $D$. The minimum size of a dominating set of $Q_{m \times n}$ is the domination number, denoted by $\gamma(Q_{m \times n})$. Values of $\gamma(Q_{m \times n}), \, 4 \leq m \leq n \leq 18,\,$ are given here, in each case with a file of minimum dominating sets (often all of them, up to symmetry) in an online appendix. In these ranges for $m$ and $n$, monotonicity fails once: $\gamma(Q_{8\times 11}) = 6 > 5 = \gamma(Q_{9 \times 11}) = \gamma(Q_{10 \times 11}) = \gamma(Q_{11 \times 11})$. Let $g(m)$ [respectively $g^{*}(m)$] be the largest integer such that $m$ queens suffice to dominate the $(m+1) \times g(m)$ board [respectively, to dominate the $(m+1) \times g^{*}(m)$ board with no two queens in a row]. Starting from the elementary bound $g(m) \leq 3m$, domination when the board is far from square is investigated. It is shown (Theorem 2) that $g(m) = 3m$ can only occur when $m \equiv 0, 1, 2, 3, \mbox{or } 4 \mbox{ (mod 9)}$, with an online appendix showing that this does occur for $m \leq 40, m \neq 3$. Also (Theorem 4), if $m \equiv 5, 6, \mbox{or } 7 \mbox{ (mod 9)}$ then $g^{*}(m) \leq 3m-2$, and if $m \equiv 8 \mbox{ (mod 9)}$ then $g^{*}(m) \leq 3m-4$. It is shown that equality holds in these bounds for $m \leq 40 $. Lower bounds on $\gamma(Q_{m \times n})$ are given. In particular, if $m \leq n$ then $\gamma(Q_{m \times n}) \geq \min \{ m,\lceil (m+n-2)/4 \rceil \}$. Two types of dominating sets (orthodox covers and centrally strong sets) are developed; each type is shown to give good upper bounds of $\gamma(Q_{m \times n})$ in several cases. Three questions are posed: whether monotonicity of $\gamma(Q_{m \times n})$ holds (other than from $(m, n) = (8, 11)$ to $(9, 11)$), whether $\gamma(Q_{m \times n}) = (m+n-2)/4$ occurs with $m \leq n < 3m+2$ (other than for $(m, n) = (3, 3)$ and $(11, 11)$), and whether the lower bound given above can be improved. A set of squares is independent if no two of its squares are adjacent. The minimum size of an independent dominating set of $Q_{m \times n}$ is the independent domination number, denoted by $i(Q_{m \times n})$. Values of $i(Q_{m \times n}), \, 4 \leq m \leq n \leq 18, \,$ are given here, in each case with some minimum dominating sets. In these ranges for $m$ and $n$, monotonicity fails twice: $i(Q_{8\times 11}) = 6 > 5 = i(Q_{9 \times 11}) = i(Q_{10 \times 11}) = i(Q_{11 \times 11})$, and $i(Q_{11 \times 18}) = 9 > 8 = i(Q_{12\times 18})$.

scholarly journals Bounds for Identifying Codes in Terms of Degree Parameters

10.37236/2036 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Florent Foucaud ◽  
Guillem Perarnau
Keyword(s):  
Large Class ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Clique Number ◽  
Minimum Size ◽  
Regular Graphs ◽  
Identifying Codes ◽  
Identifying Code ◽  
Sharp Estimates

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If $\gamma^{\text{ID}}(G)$ denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{d}+c$. We use probabilistic tools to show that for any $d\geq 3$, $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d)}$ holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove $\gamma^{\text{ID}}(G)\leq n-\tfrac{n}{\Theta(d^{3})}$. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, $\gamma^{\text{ID}}(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n$. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$.

scholarly journals New Bounds on the Double Total Domination Number of Graphs

2021 ◽  
Author(s):  
A. Cabrera-Martínez ◽  
F. A. Hernández-Mira
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

AbstractLet G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

scholarly journals General Bounds for Identifying Codes in Some Infinite Regular Graphs

10.37236/1583 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Irène Charon ◽  
Iiro Honkala ◽  
Olivier Hudry ◽  
Antoine Lobstein
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Lower And Upper Bounds ◽  
Identifying Codes ◽  
Identifying Code

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

Graph Models for Backbone Sets and Limited Packings in Networks

2021 ◽  
pp. 213-274
Author(s):  
Vadim Zverovich
Keyword(s):  
Fault Tolerant ◽  
Dominating Set ◽  
Facility Location Problem ◽  
Upper Bounds ◽  
Sensor Nodes ◽  
Theoretic Approach ◽  
Dominating Sets ◽  
Packing Number ◽  
Graph Theoretic ◽  
Threshold Functions

Here, a graph-theoretic approach is applied to some problems in networks, for example in wireless sensor networks (WSNs) where some sensor nodes should be selected to behave as a backbone/dominating set to support routing communications in an efficient and fault-tolerant way. Four different types of multiple domination (k-, k-tuple, α‎- and α‎-rate domination) are considered and recent upper bounds for cardinality of these types of dominating sets are discussed. Randomized algorithms are presented for finding multiple dominating sets whose expected size satisfies the upper bounds. Limited packings in networks are studied, in particular the k-limited packing number. One possible application of limited packings is a secure facility location problem when there is a need to place as many resources as possible in a given network subject to some security constraints. The last section is devoted to two general frameworks for multiple domination: <r,s>-domination and parametric domination. Finally, different threshold functions for multiple domination are considered.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

Independent dominating sets in graphs of girth five

2020 ◽  
pp. 1-16
Author(s):  
Ararat Harutyunyan ◽  
Paul Horn ◽  
Jacques Verstraete
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Regularity Conditions ◽  
Dominating Sets ◽  
Vertex Graph ◽  
First Results ◽  
Complete Bipartite Graphs ◽  
Image Position ◽  
Almost All ◽  
Independent Dominating Set

Abstract Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then $$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$ In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then $$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$ for some constant c independent of d. This result is sharp in the sense that as $d \rightarrow \infty$ , almost all d-regular n-vertex graphs G of girth at least five have $$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$ Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs $K_{d,d}$ , then ${\gamma_\circ}(G) = \frac{n}{2}$ . We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that ${\gamma_\circ}(G) \sim {n}/{2}$ as $d \rightarrow \infty$ . Therefore both the girth and regularity conditions are required for the main result.

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

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